×

zbMATH — the first resource for mathematics

LMI characterization of the strong delay-independent stability of linear delay systems via quadratic Lyapunov–Krasovskii functionals. (English) Zbl 0974.93060
Summary: The author proposes an analogue for linear delay systems of the characterization of asymptotic stability of rational systems by the solvability of an associated Lyapunov equation. It is shown that strong delay-independent stability of a delay system is equivalent to the feasibility of a certain linear matrix inequality (LMI), related to quadratic Lyapunov-Krasovskij functionals.

MSC:
93D20 Asymptotic stability in control theory
93C23 Control/observation systems governed by functional-differential equations
15A39 Linear inequalities of matrices
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agathoklis, P.; Foda, S., Stability and the matrix Lyapunov equation for delay differential systems, Int. J. control, 49, 2, 417-432, (1989) · Zbl 0664.93064
[2] Barnea, D.I., A method and new results for stability and instability of autonomous functional differential equations, SIAM J. appl. math., 17, 681-697, (1969) · Zbl 0181.10102
[3] P.-A. Bliman, Stability of nonlinear delay systems: delay-independent small gain theorem and frequency domain interpretation of the Lyapunov-Krasovskii method, Int. J. Control, accepted for publication. · Zbl 1015.93054
[4] P.-A. Bliman, Lyapunov-Krasovskii method and strong delay-independent stability of linear delay systems, Proceedings of the Second IFAC Workshop on Linear Time Delay Systems, Ancona, Italy, 2000, pp. 5-9.
[5] Chen, J.; Latchman, H.A., Frequency sweeping tests for stability independent of delay, IEEE trans. automat. control, 40, 9, 1640-1645, (1995) · Zbl 0834.93044
[6] Gu, K., Discretized LMI set in the stability problem of linear uncertain time-delay systems, Int. J. control, 68, 4, 923-934, (1997) · Zbl 0986.93061
[7] Hale, J.K., Theory of functional differential equations, applied mathematical sciences, vol. 3, (1977), Springer New York · Zbl 0425.34048
[8] Hale, J.K.; Infante, E.F.; Tsen, F.S.P., Stability in linear delay equations, J. math. anal. appl., 115, 533-555, (1985) · Zbl 0569.34061
[9] Hertz, D.; Jury, E.I.; Zeheb, E., Stability independent and dependent of delay for delay differential systems, J. franklin institute, 318, 3, 143-150, (1984) · Zbl 0552.34066
[10] Infante, E.F.; Castelan, W.B., A Liapunov functional for a matrix difference-differential equation, J. differential equations, 29, 439-451, (1978) · Zbl 0354.34049
[11] Kamen, E.W., On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations, IEEE trans. automat. control, 25, 5, 983-984, (1980) · Zbl 0458.93046
[12] Kamen, E.W., Linear systems with commensurate time delays: stability and stabilization independent of delay, IEEE trans. automat. control, 27, 2, 367-375, (1982) · Zbl 0517.93047
[13] Kamen, E.W., Correction to “linear systems with commensurate time delays: stability and stabilization independent of delay”, IEEE trans. automat. control, 28, 2, 248-249, (1983)
[14] V.L. Kharitonov, Robust stability analysis of time delay systems: a survey, Proc. IFAC Syst. Struct. Contr., 1998.
[15] Krasovskii, N.N., Stability of motion, (1963), Stanford University Press Stanford · Zbl 0109.06001
[16] S.-I. Niculescu, J.-M. Dion, L. Dugard, H. Li, Asymptotic stability sets for linear systems with commensurable delays: a matrix pencil approach, IEEE/IMACS CESA’96, Lille, France, 1996.
[17] S.-I. Niculescu, E.I. Verriest, L. Dugard, J.-M. Dion, Stability and robust stability of time-delay systems: a guided tour, in: Stability and control of time-delay systems, Lecture Notes in Control and Information Sciences, Vol. 228, Springer, London, 1998, pp. 1-71. · Zbl 0914.93002
[18] Rantzer, A., On the Kalman-yakubovich-Popov lemma, Syst. contr. lett., 28, 1, 7-10, (1996) · Zbl 0866.93052
[19] E.I. Verriest, A.F. Ivanov, Robust stabilization of systems with delayed feedback, Proceedings of the Second International Symposium on Implicit and Robust Systems, Warszaw, Poland, 1991, pp. 190-193.
[20] Whenzhang, H., Generalization of Liapunov’s theorem in a linear delay system, J. math. anal. appl., 142, 83-94, (1989)
[21] Zhou, K.; J.C. Doyle, with; Glover, K., Robust and optimal control, (1996), Prentice-Hall Englewood Cliffs, NJ
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.